- Open Access
Existence of periodic solutions for a type of linear difference equations with distributed delay
© Alzabut; licensee Springer. 2012
- Received: 8 February 2012
- Accepted: 3 May 2012
- Published: 3 May 2012
By employing primary algebraic techniques, we establish a necessary and sufficient condition for the existence of periodic solutions for a type of linear difference equations with distributed delay of the form
Our approach is based on constructing an adjoint equation for (*) and proving that (*) and its adjoint equation have the same number of linearly independent periodic solutions.
AMS Subject Classification: 39A11.
- difference equations
- periodic solutions
- distributed delay
Let ℕ, ℤ, ℝ be the sets of natural, integer and real numbers, respectively. By ℝ m , we denote the m-dimensional Euclidean space with elements x = col(x1, x2, . . . , x m ).
for all periodic solutions y(t) of period ω of the adjoint equation y'(t) = - A T (t)y(t), where A ∈ C(ℝ, ℝm×m) and f ∈ C(ℝ, ℝ m ) are periodic functions of period ω; see for instance . By "T", we mean the transposition.
The same problem has been investigated for linear impulsive delay differential equations [3, 4]. The discrete analog of the above mentioned result has been recently studied in . We suggest the reader to consult [6–10] for more results regarding existence of periodic solutions for difference equations.
ζ(n, k) is normalized so that ζ(n, s) = 0 for s ≥ -1 and for s ≤ - d + 1 where d > 3 is a positive integer;
There exists a positive real number γ such that .
has a unique solution x(n) which is defined for n ∈ ℕ(n0 - d + 1) and satisfies the initial conditions (5). To emphasize the dependence of the solution on the initial point n0 and the initial functions ϕ, we may use the notation x(n) = x(n; n0, ϕ).
Our approach is based on constructing an adjoint equation for (4) with respect to a discrete analog for function (3) and proving that (4) and its adjoint equation have the same number of linearly independent periodic solutions. We shall employ some primary algebraic techniques to prove the main results of this article. It is worth mentioning here that the equation under consideration in this article (Equation (4)) is given in general form so it includes many particular cases of difference equations with pure delays; see [5, 11–13] for more details.
This section is devoted to certain auxiliary assertions that will be needed in the proof of the main theorem. Lemma 2.1 which introduces the main result of this section is needed to define an adjoint equation for (4). Lemmas 2.4 and 2.7 give representations of solutions of the considered equations. The proof of these lemmas were given in . For the benefit of the readers, however, we state these lemmas along with their proofs.
is an adjoint equation of (4) with respect to (6). The following lemma proves meaningful.
where < ·,· > is defined by (6).
By changing the indices and using the properties of ζ, we see that the above equation is equal to zero. The proof is finished.
Remark 2.2 In virtue of Lemma 2.1, we may say that Equation (7) is an adjoint of (4). It is easy to verify also that the adjoint of (7) is (4), i.e., they are mutually adjoint of each other.
where f is a sequence with values in ℝ m .
Definition 2.3 A matrix solution X(n, α) of (4) satisfying X(α, α) = I, (I is an identity matrix), and X(n, α) = 0 for n < α is called a fundamental function of (4).
Definition 2.6 A matrix solution Y (n, α) of (7) satisfying Y (α, α) = I and Y (n, α) = 0 for n > α is called a fundamental function of (7).
Upon using the properties of the fundamental functions X(n, n0) and Y (n, n0), identity (16) is obtained.
Remark 2.9 Formulas (14) and (15) can be derived from function (6). Indeed, replacing X by x or Y by y in (17), using (16) and employing the properties of X and Y we obtain the desired results.
ζ(n, k): ℕ × ℤ → ℝm × mis p periodic sequence in n, p > d;
f: ℕ → ℝ m is p a periodic sequence, p > d.
Let x(n) = x(n; φ) be the solution of Equation (11) defined for n ≥ 1 such that x(n) coincides with φ on [-d + 2, 2]. The periodicity of the equation implies that x(n + p; φ) is likewise a solution of the equation defined for n + p ≥ d. If this solution coincides with φ in [-d +2, 2], then on the basis of the uniqueness theorem it follows that x(n + p; φ) = x(n; φ) for all n ≥ -d + 2 and the solution is periodic. Thus the periodicity condition of the solution is written as x(n + p; φ) = φ(n) for n ∈ [-d + 2, 2]. If W is defined by Wφ = x(n + p; φ), n ∈ [-d + 2, 2], then it follows that x(n) is periodic if and only if Wφ = φ, i.e., φ is a fixed point of W.
for matrix sequences Ψ and Φ defined on [-d + 2, 2] as long as multiplication is possible. Note that < Ψ(s), Φ(s) > could be either a number or a vector or a matrix, depending on the sizes of Ψ and Φ.
The following lemma, which is a discrete analogue of [4, Lemma 4], plays a key role in our later analysis. Its proof is straightforward and can be achieved directly by changing the order of summations.
where , s ∈ [-d + 2, 2]. The right side of the above equation is nothing but . Thus the eigenvalues of the operators and coincide and in addition, if ψ is an eigenfunction for , then is an eigenfunction for .
Lemma 3.2 Equations (4) and (7) have the same number of linearly independent periodic solutions of period p > d.
is a solution of (22).
where , and , a k (α) >. In view of (30), (31)
have the same number of linearly independent solutions. To a solution of (40) corresponds and to this corresponds the solution φ(s) = v(s) + < Г T (s, α), v(α) > for the equation ρφ(s) - Uφ(s) = 0, linearly independent solutions corresponding to the linearly independent solutions of Equation (40). Likewisely, a solution of the equation will correspond to a solution of Equation (37) which coincides with (41), linearly independent solutions corresponding to linearly independent solutions. It follows from here that the equations ρφ(s) - Uφ(s) = 0 and have the same number of independent solutions, which implies in particular the fact that U and have the same eigenvalues, hence if ρ is a multiplier of the equation, is a multiplier of the adjoint equation. The proof of Lemma 3.2 is completed.
We are now in a position to state and prove the main result of this article.
for all periodic solutions y(n) of period p of the adjoint Equation (7).
which is the same as (42).
which is clearly zero by our assumption (42). The proof is finished.
which is equal to zero for any periodic solution y of Equation (50) under the initial condition y(1) - y(3) = 0. By the result of Theorem 3:3, we conclude that there exist periodic solutions of period 4 for Equation (49).
The author would like to express his sincere thanks for the valuable comments of the reviewers which improved the exposition of the article.
- Miller RK, Michal AN: Ordinary Differential Equations. Academic Press, New York; 1982.Google Scholar
- Halanay A: Differential Equations: Stability, Oscillation, Time Lags. Academic Press, New York; 1966.Google Scholar
- Alzabut JO: A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Discr Contin Dyn Syst Supplement 2007, 35–43.Google Scholar
- Akhmet MU, Alzabut JO, Zafer A: On periodic solutions of linear impulsive delay differential systems. Dyn Contin Discr Impuls Syst Ser A Math Anal 2008, 15(5):621–631.MathSciNetGoogle Scholar
- Alzabut JO: On existence of periodic solutions for a class of linear delay difference equations. Commun Appl Anal 2010, 14(2):191–202.MathSciNetGoogle Scholar
- Dannan F, Elaydi S, Liu P: Periodic solutions of difference equations. J Diff Equ Appl 2000, 6(2):203–232. 10.1080/10236190008808222MathSciNetView ArticleGoogle Scholar
- Elaydi S, Zhang S: Stability and periodicity of difference equations with finite delay. Funkcial Ekvac 1994, 37(3):401–413.MathSciNetGoogle Scholar
- Cai X, Yu J: Existence theorems of periodic solutions for second-order nonlinear difference equations. Adv Diff Equ 2008., 11: (Article ID 247071) (2008). doi:10.1155/2008/247071Google Scholar
- Vidal C: Existence of periodic and almost periodic solutions of abstract retarded functional difference equations in phase spaces. Adv Diff Equ 2009., 19: (Article ID 380568) (2009). doi:10.1155/2009/380568Google Scholar
- Song Y: Periodic and almost periodic solutions of functional difference equations with finite delay. Adv Diff Equ 2007., 15: (Article ID 68023)(2007). doi:10.1155/2007/68023Google Scholar
- Agarwal RP, Wong PJY: Advanced Topics in Difference Equations. Kluwer, Dordrecht 1997.Google Scholar
- Agarwal RP: Difference Equations and Inequalities, Theory, Methods and Applications. 2nd edition. Marcel Dekker, New York; 2000.Google Scholar
- Elayadi S: An Introduction to Difference Equations. 3rd edition. Springer, New York; 2005.Google Scholar
- Alzabut JO, Abdeljawad T: Perron-type criterion for linear difference equations with distributed delay. Discr Dyn Natl Soc 2007., 12: (Article ID 10840) (2007). doi:10.1155/2007/10840Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.